Full Waveform Inversion (FWI) updates an earth model by minimizing the difference between modeled and observed seismic data. The significant cost of computing the FWI remains a primary practical limitation, especially in industry applications. This significant cost is driven by determining the solution of the wave equation for every source in the acquired seismic data for a given earth model. This solution is repeated multiple times, because the earth model is updated in an iterative manner.
In general, a solution of the wave equation from the source is referred to as a “forward modeling”. Similarly, a solution of the wave equation from the receivers is referred to as a “reverse modeling”. An update to the current earth model is obtained by combining the forward and reverse modelings for a given source and receiver acquisition geometry to yield a “gradient” update to the current earth model, i.e., a direction in which to make a perturbation in the earth parameters of the earth model.
The typical workflow in a “sequential source” FWI implementation includes using the current earth model to perform an individual forward modeling for each source in a survey to generate a synthetic data set for each source. A difference is determined between each synthetic data set and a field data set of actual data at each given receiver in the set of receivers to produce a residual data set. A reverse modeling is then performed for the set of data residuals. This yields a “local” contribution to the gradient for the given source. This is repeated for each source. Not every source needs to be used in this update; however, the sources are modeled as individual entities. The “local” gradient contributions from each of these individual sources are combined to give the “total” gradient for this iteration. The “total” gradient is used to determine a direction of a perturbation in the earth parameters of the current earth model, and the current earth model is updated based on the “total” gradient.
These steps are performed iteratively for a given seismic bandpass frequency or frequency block. For example, these steps are repeated until convergence, or until a given maximum number of iterations is achieved. This iterative process to achieve convergence or to conduct a given maximum number of iterations are repeated at various seismic bandpass frequencies, updating the earth model as the process proceeds through the various seismic bandpass frequencies. Therefore, sequential source FWI implementation requires modeling for individual sources, in multiple iterations and multiple frequency blocks, resulting in high computational costs.